--- license: apache-2.0 language: - en size_categories: - 1K **Unofficial.** Deep Arithmark is an independent companion tier. It is not > affiliated with, endorsed by, or published by AxiomicLabs, the authors of > ArithMark 3.0, and it is not an official ArithMark release. It follows their > task contract and credits them throughout; any errors are the author's. If you > use this benchmark, cite ArithMark 3.0 as well. | | | | --- | ---: | | Examples | 1,000 | | Choices per example | 4 | | Chance | 25 % | | Depth tiers | 2, 3, 4, 5 — 250 each | | Templates / topic labels | 15 / 14 | | Mean prompt length | 29.2 words (max 43) | | Language, source | English, synthetic (deterministic seed) | ## Quick start ```python from datasets import load_dataset ds = load_dataset("OPENGCM/DeepArithmark-1", split="test") item = ds[0] # score the four continuations of item["ctx"] by length-normalised # log-likelihood, take the argmax, compare with int(item["label"]) ``` Score models directly, including an ArithMark 3.0 comparison on the same weights: ```bash python bench_deeparithmark.py --compare --model HuggingFaceTB/SmolLM2-135M ``` ArithMark 3.0's own scorer runs on this file unchanged (verified against that script's loader): ```bash python bencharithmark-3.py --data-path data/deeparithmark-1.jsonl --model gpt2 ``` ## Why it is harder ArithMark 3.0's four continuations are **adjacent integers** (43 / 44 / 45 / 46). Rejecting a wrong option only needs "the answer is about 44", so a magnitude prior is sufficient and the arithmetic is optional. Deep Arithmark keeps the contract and changes three things: 1. **Depth.** Every item needs 2–5 dependent operations; later steps consume earlier results. `depth` equals the length of `metadata.solution_steps`, and the build fails otherwise. 2. **Decoys with provenance.** No decoy is "answer ± 1". Each one is the exact value of a specific plausible mistake — the state after step 2 instead of step 5, the percentage on the wrong base, the divisor `n` instead of `n + 1`, the dropped unit conversion. `metadata.option_sources` names the mistake per ending, so error analysis can ask *which step* a model botches. 3. **Balanced on three axes.** Correct slot, the answer's *numeric rank*, and the *roundness class* of all four options are each balanced. "Pick the largest" and "pick the roundest number" are both worth 25 %. ## One item ```json { "ind": 3, "activity_label": "deeparithmark::deeparithmark-1::multi_step_inventory::depth4::hard", "ctx": "A warehouse held 420 units. A delivery added 260, then 210 units were shipped out, and 40 were found damaged. The rest were packed into crates of 16. The number of full crates packed was", "endings": [" 26 crates.", " 29 crates.", " 27 crates.", " 31 crates."], "label": "0", "source_id": "deeparithmark_v1_multi_step_inventory_depth4_0003", "split": "test", "split_type": "synthetic", "metadata": { "answer": "26", "unit": "crates", "topic": "multi_step_inventory", "depth": 4, "option_values": [26, 29, 27, 31], "option_sources": ["correct", "forgot the damaged units", "counted the loose units as a crate", "added the damaged units"], "answer_rank": 0, "solution_steps": [ "after delivery: 420 + 260 = 680", "after shipping: 680 - 210 = 470", "good units: 470 - 40 = 430", "full crates: 430 / 16 = 26 (remainder 14)" ] } } ``` The answer is ` 26 crates.` — `420 + 260 − 210 − 40 = 430`, and `430 // 16 = 26` full crates. `29` is what you get by forgetting the damaged units, `27` by counting the loose units as a crate, `31` by adding them. ## Scores Zero-shot, `acc_norm`, bf16, batch 32, `--max-context 1024`. ArithMark 3.0 numbers were produced on the same weights in the same run, not copied from a table. `soft` is the mean probability of the correct option after a softmax over the four scores; `d2`–`d5` are `acc_norm` per depth tier (250 items each, chance 25 %). Δ is Deep Arithmark minus ArithMark 3.0. Every row is in one file, `benchmarks/deeparithmark_17models.json`. `LiquidAI/LFM3-1.7B-Base` ships remote modelling code written against transformers 4.x — it decorates its convolution with a hub-kernel helper that 5.x removed — so the scorer stubs two deleted helpers and runs the checkpoint's own pure-torch path. Re-scoring it at batch 8 instead of 32 moved two items (31.90 % against 31.70 %), which is bf16 kernel noise, not a padding leak. | Model | Params | Deep | soft | d2 | d3 | d4 | d5 | ArithMark-3 | Δ | | --- | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | | SmolLM3-3B-Base (2025-06) | 3.08 B | 36.50 % | 27.09 % | 57.6 | 35.6 | 26.0 | 26.8 | 75.70 % | −39.20 | | Qwen3.5-2B-Base (2026-02) | 1.88 B | 36.00 % | 27.01 % | 59.2 | 26.4 | 28.8 | 29.6 | 86.70 % | −50.70 | | LFM3-1.7B-Base (2026-10) | 1.73 B | 31.70 % | 26.42 % | 41.2 | 38.4 | 23.6 | 23.6 | **93.10 %** | **−61.40** | | granite-4.0-1b-base (2025-10) | 1.63 B | 29.90 % | 26.13 % | 39.2 | 29.2 | 28.4 | 22.8 | 84.20 % | −54.30 | | Qwen2.5-1.5B-Instruct | 1.54 B | 38.00 % | 27.23 % | 60.4 | 40.8 | 28.4 | 22.4 | — | — | | Qwen3.5-0.8B-Base | 752 M | 26.70 % | 25.48 % | 28.8 | 24.4 | 28.8 | 24.8 | 75.40 % | −48.70 | | Tev1-0.8B-experimental | 752 M | 23.10 % | 25.30 % | 20.8 | 24.4 | 23.6 | 23.6 | 71.40 % | −48.30 | | Qwen3-0.6B-Base | 596 M | **29.20 %** | 25.73 % | 37.6 | 32.0 | 28.8 | 18.4 | 68.40 % | −39.20 | | Qwen2.5-0.5B | 494 M | 24.60 % | 25.50 % | 24.8 | 26.4 | 23.6 | 23.6 | 60.90 % | −36.30 | | SmolLM2-360M-Instruct | 362 M | 23.80 % | 25.19 % | 24.8 | 26.8 | 26.4 | 17.2 | — | — | | LFM2.5-350M-Base | 354 M | 20.00 % | 24.01 % | 17.2 | 23.6 | 21.2 | 18.0 | 36.00 % | −16.00 | | granite-4.0-350m-base | 352 M | 24.10 % | 25.34 % | 24.4 | 22.0 | 23.6 | 26.4 | 75.70 % | −51.60 | | GPT-X3-150M | 149 M | 25.20 % | 25.07 % | 23.6 | 24.0 | 25.6 | 27.6 | 49.60 % | −24.40 | | SmolLM2-135M | 135 M | 24.80 % | 25.21 % | 18.0 | 29.2 | 26.4 | 25.6 | 38.60 % | −13.80 | | gpt2 | 124 M | 24.00 % | 24.98 % | 22.0 | 23.2 | 25.2 | 25.6 | 34.70 % | −10.70 | | nanowhale-100m-base | 110 M | 21.30 % | 24.46 % | 15.6 | 22.8 | 23.6 | 23.2 | 26.50 % | −5.20 | | EleutherAI/pythia-70m | 70 M | 22.70 % | 24.34 % | 21.6 | 26.0 | 21.6 | 21.6 | 28.80 % | −6.10 | `Tev1-0.8B-experimental` is a post-trained derivative of `Qwen/Qwen3.5-0.8B` (identical parameter count), so that pair is base vs. post-trained on one architecture rather than two models. `gemma-3-270m` and `MobileLLM-R1-950M` are gated and have not been scored. **Four bands.** Below ~350 M nothing clears chance, and two models sit significantly *below* it — nanowhale-100M (21.3 %, p = 0.007) and LFM2.5-350M (20.0 %, p = 0.0002) — meaning a stable losing strategy rather than absent arithmetic encoding. **Qwen3-0.6B-Base is the first sub-1B model to clear chance significantly: 29.20 %, p = 0.0027.** The 1.6–3.1 B band clears it comfortably and then stalls at 29.9–36.5 % (p = 4.5e−4 to 8e−16). Only the post-trained Qwen2.5-1.5B-Instruct declines smoothly all the way down (60.4 → 40.8 → 28.4 → 22.4); every *base* model from 1.6 B to 3.1 B is back at chance by depth 4. **The one to watch is Qwen3-0.6B's depth curve:** 37.6 → 32.0 → 28.8 → **18.4 %**, significant at both ends (depth 2 p = 1e-5, depth 5 p = 0.016). Being *below* chance at depth 5 means it commits to a stable wrong path — it starts computing and drops the last step — rather than guessing. Option rank is balanced uniformly across depth, so no rank preference can produce that shape. **Hardness scales with how well a model exploits ±1 options.** granite-4.0-350M Base scores **75.70 %** on ArithMark 3.0, the highest sub-1B result here, and 24.10 % on this dataset — a **51.6-point** drop, against GPT-X3-150M's 24.4. **The extreme case is now LFM3-1.7B-Base.** It scores **93.10 %** on ArithMark 3.0 — the highest figure in this table — and 31.70 % here: a **61.4-point** drop, larger than granite-4.0-350M's 51.6. It is also the only model whose depth-3 bucket stays within three points of its depth-2 bucket with both above chance (41.2 → 38.4, p = 3e−6) before falling off a cliff at depth 4 (23.6). That skill is concentrated in two templates: `share_then_split` 93.2 % and `time_clock_end` 77.3 %, with `money_discount_fee` at 3.3 %. A 1.7 B model that nearly aces another benchmark is still a single-template specialist here. **Scaling 0.6 B → 3.1 B buys about seven points.** Qwen3-0.6B 29.2, granite-1B 29.9, LFM3-1.7B 31.7, Qwen3.5-2B 36.0, SmolLM3-3B 36.5: a 5× parameter increase for +7.3 points, while ArithMark 3.0 on the same runs spans 68.4 → 93.1. The aggregate also hides opposite failure shapes at the same level. Qwen3.5-2B has the best depth-2 bucket in the table (59.2 %, p = 2e−30) and collapses at depth 3 (26.4 %); SmolLM3-3B is weaker at depth 2 (57.6) yet still holds depth 3 (35.6 %, p = 2e−4). Same ~36 %, different structure — which is why the curve is the reportable unit, not the single number. **A chance score is not always guessing.** GPT-X3-150M scores 49.60 % on ArithMark 3.0 and 25.20 % here, but its per-template accuracy spans 3.0 % to 54.5 % (χ² = 63.2, df = 14, p < 1e-4): a small-number prior (picks the smallest option 34.8 % of the time, the largest 15.8 %) plus one stable wrong path per family. These cancel to chance in aggregate, so treat per-template rows as preference, not partial skill. Worked through in the design notes, §6. For a sub-1B leaderboard, report the per-depth curve rather than only the single number. The two solvable templates are stable across families: at 1.5B-instruct `share_then_split` (86.4 %) and `time_clock_end` (85.9 %), at 1.7 B `share_then_split` 93.2 %, and at 2 B `time_clock_end` 90.6 %. ## Composition | Depth | Examples | Typical item | | ---: | ---: | --- | | 2 | 250 | percent off a price, then a fee | | 3 | 250 | unit conversion, add, subtract | | 4 | 250 | receive / ship / damage / pack | | 5 | 250 | two percentage days, then palletise | | Template | Depth | Examples | Template | Depth | Examples | | --- | ---: | ---: | --- | ---: | ---: | | `money_discount_fee` | 2 | 122 | `schedule_sessions` | 3 | 47 | | `time_clock_end` | 2 | 128 | `inventory_boxes` | 4 | 62 | | `measure_cm_cut` | 3 | 33 | `mix_split` | 4 | 63 | | `share_then_split` | 3 | 44 | `tank_fill` | 4 | 62 | | `two_rate_pay` | 3 | 45 | `discount_tax_coupon_split` | 4 | 63 | | `stock_percent_pack` | 3 | 48 | `salary_overtime_tax` | 5 | 84 | | `trip_minutes` | 3 | 33 | `inventory_percent_chain` | 5 | 83 | | | | | `prize_ratio_charity` | 5 | 83 | Correct slot is 252 / 252 / 248 / 248, answer numeric rank (smallest → largest) is 248 / 252 / 248 / 252, and the `slot × rank × depth` grid holds 15–16 items per cell. ### Shortcut audit No strategy that skips the arithmetic beats chance (`python deeparithmark.py audit`, chance 25.00 %): | Strategy | Score | | --- | ---: | | pick largest / smallest value | 25.20 % / 24.80 % | | pick the roundest number | 25.20 % | | fixed slot 0 / 1 / 2 / 3 | 25.20 / 25.20 / 24.80 / 24.80 % | | longest / shortest text | 25.50 % / 24.00 % | | char-3-gram classifier over option text | 25.47 % | This is a build constraint, not a happy accident. An early build scored gpt2 **below** chance (21.1 %) because several decoys were structurally rounder than the answer, making a roundness prior anti-correlated with correctness. All four options are now forced into the same roundness class. ## Limitations - 1,000 items → roughly ±2.7 points (95 % CI) at chance. Differences under ~4 points are noise. - Hard 4-way accuracy gives no partial credit by design; use the soft score for a gradient. - Single `test` split, no train split (evaluation set). - 15 templates only: fine-tuning on them would overfit. - Rank balance is global, not per-template, so per-template rows read as preference rather than skill. - Clock items keep all four options inside one half of the day, so no noon rollover is exercised. ## How it was generated `deeparithmark.py build` renders the file deterministically from seed `20261027`: 15 parameterised families, exact integer arithmetic, one decoy pool per family. `python deeparithmark.py selftest` re-checks every claim on this page — schema, unit consistency, option distinctness, step count vs depth, all three balance axes, and the audit above. The seed makes the *instances* reproducible and unseen; it does not make the template shapes novel. Do not train on this file. ## Licence and attribution Released under the **Apache License 2.0**. Dataset SHA-256: `8a54eb90b66b142b7fea422b34d851746d1cabc476dcba906c7f3ad5ddc66ee0`. **Upstream work.** Deep Arithmark follows [ArithMark 3.0](https://huggingface.co/datasets/AxiomicLabs/Arithmark-3.0) by AxiomicLabs, also Apache-2.0. Reused: the *task contract* — one unfinished sentence, four continuations with a leading space, a single correct index, and scoring by length-normalised continuation log-likelihood — plus the JSONL schema, so their scorer runs on this file unchanged. **What is not reused.** No ArithMark 3.0 item appears here. Every prompt comes from the parameterised families in `deeparithmark.py`, and the released file shares **0 identical contexts and 0 eight-word n-grams** with the 1,000 upstream items. The scorer is an independent implementation of the same protocol, not copied code. **Trademarks.** "ArithMark" and "AxiomicLabs" remain the property of their owner. Their use here is descriptive, to state which benchmark this one extends. It does not imply affiliation or endorsement — see the notice at the top. ## Files | File | What it is | | --- | --- | | `data/deeparithmark-1.jsonl` | The dataset — 1,000 items, one JSON object per line | | `deeparithmark.py` | Deterministic generator, plus `selftest` and the shortcut `audit` | | `bench_deeparithmark.py` | Scorer with per-depth and per-template breakdowns and an ArithMark 3.0 comparison | | `DEEPARITHMARK_DESIGN.md` | Design rationale, fairness audit, measurement discussion | | `benchmarks/deeparithmark_6models.json` | Raw results for the 2026 sub-1B sweep |